The performance of both the original and enhanced STROGANOFF was evaluated on the sunspots data of
Table 4.5 and is shown in
Table 4.17. As expected, the enhanced implementation is considerably better than the original STROGANOFF. Note, however, that in the GEP-ESM experiment a three-genic system was used and, therefore, the system we are simulating does not correspond to the enhanced STROGANOFF as described by
Nikolaev and Iba (2001) but, rather, is a much more efficient algorithm, as it benefits from the multigenic nature of gene expression programming. Indeed, the multigenic system works considerably better than the unigenic one. It is worth emphasizing that the implementation of multiple parse trees in GP is unfeasible and so is a system similar to the one used in the GEP-ESM experiment. And, of course, the facility for the manipulation of random constants in GP is much less versatile than the one used in GEP and, in fact, the coefficients in GP can only be discovered a posteriori, usually by a neural network. This obviously raises the question of what is exactly the GP doing in this case for, without the coefficients, the polynomials are useless (see
Table 4.18 below).
Table 4.17
Settings used in the GEP simulation of the original STROGANOFF (GEP-OS) and the enhanced STROGANOFF using a unigenic system
(GEP-ESU) and a multigenic system (GEP-ESM).
| |
GEP-OS |
GEP-ESU |
GEP-ESM |
| Number
of runs |
100 |
100 |
100 |
| Number
of generations |
5000 |
5000 |
5000 |
| Population
size |
100 |
100 |
100 |
| Number
of fitness cases |
90 |
90 |
90 |
| Function
set |
(F9)16 |
F1 -
F16 |
F1 -
F16 |
| Terminal
set |
d0 -
d9 |
d0 -
d9 |
d0 -
d9 |
| Random
constants array length |
120 |
120 |
40 |
| Random
constants range |
[-1,1] |
[-1,1] |
[-1,1] |
| Head
length |
21 |
21 |
7 |
| Number
of genes |
1 |
1 |
3 |
| Linking
function |
-- |
-- |
+ |
| Chromosome
length |
169 |
169 |
171 |
| Head/tail
mutation rate |
0.044 |
0.044 |
0.044 |
| Dc
mutation rate |
0.06 |
0.06 |
0.06 |
| One-point
recombination rate |
0.3 |
0.3 |
0.3 |
| Two-point
recombination rate |
0.3 |
0.3 |
0.3 |
| Gene
recombination rate |
-- |
-- |
0.1 |
| IS
transposition rate |
0.1 |
0.1 |
0.1 |
| IS
elements length |
1,2,3 |
1,2,3 |
1,2,3 |
| RIS
transposition rate |
0.1 |
0.1 |
0.1 |
| RIS
elements length |
1,2,3 |
1,2,3 |
1,2,3 |
| Gene
transposition rate |
-- |
-- |
0.1 |
| Random
constants mutation rate |
0.25 |
0.25 |
0.25 |
| Dc
specific transposition rate |
0.1 |
0.1 |
0.1 |
| Dc
specific IS elements length |
5,7,9 |
5,7,9 |
5,7,9 |
| Selection
range |
1000% |
1000% |
1000% |
| Precision |
0% |
0% |
0% |
| Average
best-of-run fitness |
86069.183 |
86566.298 |
86881.997 |
| Average
best-of-run R-square |
0.4949506 |
0.6712016 |
0.7631128 |
Indeed, continuing our discussion about the importance of random constants in evolutionary symbolic regression, there is a simple experiment we could do. We could implement the bivariate polynomials presented in
Table 4.16 and try to evolve complex polynomial models with them
(Table 4.18). On the one hand, the comparison of this experiment with the experiments summarized in
Table 4.17 clearly show that coefficients are indeed fundamental to the evolution of polynomials. But genuine polynomials with coefficients are much less efficient than a simple GEA (compare
Table 4.10 with
Table 4.17) and, therefore, the role of numerical constants in evolutionary symbolic regression is a marginal one and has only theoretical interest.
Table 4.18
The role of coefficients in polynomial evolution.
| Number
of runs |
100 |
| Number
of generations |
5000 |
| Population
size |
100 |
| Number
of fitness cases |
90 |
| Function
set |
F1 -
F16 |
| Terminal
set |
d0 -
d9 |
| Head
length |
7 |
| Number
of genes |
3 |
| Linking
function |
+ |
| Chromosome
length |
45 |
| Mutation
rate |
0.044 |
| One-point
recombination rate |
0.3 |
| Two-point
recombination rate |
0.3 |
| Gene
recombination rate |
0.1 |
| IS
transposition rate |
0.1 |
| IS
elements length |
1,2,3 |
| RIS
transposition rate |
0.1 |
| RIS
elements length |
1,2,3 |
| Gene
transposition rate |
0.1 |
| Selection
range |
1000% |
| Precision |
0% |
| Average
best-of-run fitness |
73218 |
Several conclusions can be drawn from the experiments presented here. First, a STROGANOFF-like system exploring second-order bivariate basis polynomials, although mathematically appealing, is extremely inefficient in evolutionary terms. Not only is its performance significantly worse but also its structural complexity is considerably more complicated. Second, the finding of the coefficients is, as expected, fundamental to the construction of models based on high-order multivariate polynomials. And, finally, a simple GEP system with the usual set of mathematical functions is much more efficient than the complicated and computationally expensive STROGANOFF systems.
In the next section we are going to use the settings of the winner algorithm (the simple GEA with the basic arithmetic functions) to evolve a model to make predictions about sunspots.
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